Finite Group Databasebeta

← #128109

Subgroups — #128109

Order 512.

718 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21742C211yesmaximal, characteristic
2#20722C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#7320C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6#2920—21no—
7C2×#407C411yescharacteristic, complemented
8C2×#407C411yescharacteristic, complemented
9#2280—21no—
10#1870—21no—
11#1870—21no—
12#1870C412yescomplemented
13#1870C2212yes—
14#1819—21no—
15#1817—21no—
16#1799—21no—
17#1799C2212yes—
18#1781C412yes—
19#1779—21no—
20C4×#99—21no—
21C4×#99C412yes—
22#1466—21no—
23#1466C2212yes—
24#1462—21no—
25#1462C412yescomplemented
26#1460—21no—
27#1460C412yescomplemented
28C23×D4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C22×M16C2×C412yes—
33C2×C4×D4—41no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4—41no—
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4D411yescharacteristic
43C2×C4×D4Q811yescharacteristic
44C2×C4×D4C2311yescharacteristic
45C2×#117—41no—
46C22×C4⋊C4Q811yescharacteristic
47C22×C22⋊C4C2×C411yescharacteristic
48C22×C22⋊C4D411yescharacteristic
49C22×C42D411yescharacteristic
50#432—22no—
51#432—41no—
52#431—22no—
53#431—41no—
54#423—41no—
55C2×#99—22no—
56C2×#99C2×C414yes—
57#407—22no—
58#407—41no—
59#407—41no—
60#407—41no—
61#407—41no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8C2×C412yes—
68C4×M16—41no—
69#381—41no—
70#380—22no—
71#380—41no—
72#379—22no—
73C4×C22⋊C4—22no—
74C4×C22⋊C4—41no—
75#342—22no—
76#342—22no—
77#336—22no—
78#336—22no—
79#323—22no—
80#323—22no—
81#323—22no—
82#323C2×C414yes—
83C25C4211yescharacteristic
84C25C22⋊C411yescharacteristic
85C22×D4—42no—
86C22×D4—41no—
87C22×D4—41no—
88C22×D4—42no—
89C22×D4C4211yescharacteristic
90C22×D4C22⋊C411yescharacteristic
91C22×D4C4⋊C411yescharacteristic
92C22×D4C22×C411yescharacteristic
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—82no—
102C23×C4—82no—
103C23×C4C4211yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C4⋊C411yescharacteristic
110C23×C4C4⋊C411yescharacteristic
111C23×C4C4⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C22×C411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C2×M16—42no—
117C2×M16—42no—
118C2×M16—42no—
119C2×M16—82no—
120C2×M16—82no—
121C22×C8—22no—
122C22×C8—22no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—41no—
129C4×D4—41no—
130C4×D4—41no—
131C4×D4—41no—
132C4×D4—41no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4—41no—
136C4×D4—41no—
137C4×D4—41no—
138C4×D4—41no—
139#117—84no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4C4⋊C411yescharacteristic
144C2×C4⋊C4C4⋊C411yescharacteristic
145C2×C4⋊C4C4⋊C411yescharacteristic
146C2×C4⋊C4C2×Q811yescharacteristic
147C2×C22⋊C4—24no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4C4211yescharacteristic
155C2×C22⋊C4C22⋊C411yescharacteristic
156C2×C22⋊C4C22⋊C411yescharacteristic
157C2×C22⋊C4C22⋊C411yescharacteristic
158C2×C22⋊C4C4⋊C411yescharacteristic
159C2×C22⋊C4C4⋊C411yescharacteristic
160C2×C22⋊C4C22×C411yescharacteristic
161C2×C22⋊C4C2×D411yescharacteristic
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—42no—
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C4⋊C411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C4⋊C8—42no—
172C4⋊C8—42no—
173#99—44no—
174#99—44no—
175C22⋊C8—24no—
176C22⋊C8—42no—
177C22⋊C8—42no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—42no—
183C22⋊C8—82no—
184C22⋊C8—82no—
185#97—42no—
186C4×C8—42no—
187#95—42no—
188#95—42no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—22no—
194C24—41no—
195C24—41no—
196C24—41no—
197C24—81no—
198C24—81no—
199C24—81no—
200C24—81no—
201C24#9511yescharacteristic
202C24#9511yescharacteristic
203C24#9511yescharacteristic
204C24C2×C4211yescharacteristic
205C24C2×C22⋊C411yescharacteristic
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—41no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—48no—
219C2×D4—48no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—81no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4#9511yescharacteristic
268C22×C4#9511yescharacteristic
269C22×C4#9511yescharacteristic
270C22×C4#9511yescharacteristic
271C22×C4#9511yescharacteristic
272C22×C4#9911yescharacteristic
273C22×C4#9911yescharacteristic
274C22×C4#9911yescharacteristic
275C22×C4#9911yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C4⋊C411yescharacteristic
279C22×C4C2×C4⋊C411yescharacteristic
280C22×C4C2×C4⋊C411yescharacteristic
281M16—84no—
282M16—84no—
283M16—84no—
284C2×C8—44no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—84no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—84no—
307C22⋊C4—84no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—84no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—84no—
353C23—84no—
354C23—84no—
355C23—84no—
356C23#34211yescharacteristic
357C23#34211yescharacteristic
358C23C2×#9511yescharacteristic
359C23#41312yes—
360D4—44no—
361D4—44no—
362D4—44no—
363D4—44no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4#34211yescharacteristic
415C2×C4#34211yescharacteristic
416C2×C4#34412yes—
417C2×C4C2×#9911yescharacteristic
418C2×C4C2×#9911yescharacteristic
419C8—82no—
420C8—84no—
421C8—82no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—22no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—84no—
453C22—84no—
454C22—84no—
455C22—84no—
456C22C2×#34211yescharacteristic
457C22#174211yescharacteristic
458C22C2×#34411yescharacteristic
459C4—21no—
460C4—21no—
461C4—21no—
462C4—22no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—164no—
471C4#212112yes—
472C2—21no—
473C2—21no—
474C2—42no—
475C2—41no—
476C2—41no—
477C2—81no—
478C2—81no—
479C2—81no—
480C2#810411yescharacteristic
481C2#1257411yescharacteristic
482C2#1257711yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC24C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7320C2semidirect
7. C2×#407C4semidirect
8. C2×#407C4semidirect
12. #1870C4semidirect
25. #1462C4semidirect
27. #1460C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#217422
#207222
#173642
#73202